Go to Display Properties, Settings, Advanced, Monitor and set the refresh rate to 60 Hz. Select Apply, and see if the noise goes away. If it does, try increasing the refresh rate one step at a time until you find the threshold at which the noise starts.

There are many things inside a monitor that could be causing a whine. Anything as small as a whiney capacitor, or as large as the flyback transformer. The fact that it stops whining after 30 seconds or so may be of some significance. How old is the monitor?